Minimal Length Maximal Green Sequences
Alexander Garver
∗1, Thomas McConville
†2and Khrystyna Serhiyenko
‡31Laboratoire de Combinatoire et d’Informatique Mathématique, Université du Québec à Montréal
2Department of Mathematics, Massachusetts Institute of Technology
3Department of Mathematics, University of California, Berkeley
Abstract. Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. It is an open problem to determine what lengths are achieved by maximal green sequences of a quiver. We use the combinatorics of surface triangulations to address this problem. Our main result is a formula for the length of minimal length maximal green sequences of quivers defined by triangulations of an annulus or a punctured disk.
Résumé. Les suites vertes maximales sont des objets importants dans la théorie des représentation, les algèbres amassées et la théorie des cordes. C’est un problème ouvert que de déterminer les longueurs que peuvent prendre les suites vertes maximales d’un carquois. Nous utilisons la combinatoire des triangulations de surface pour étudier ce problème. Notre résultat principal est une formule pour la longueur minimale des suites vertes maximales de carquois définis par des triangulations d’un anneau ou d’un disque perforé.
Keywords: quiver mutation, maximal green sequence, triangulated surface
1 Introduction
A maximal green sequence is a distinguished sequence of local transformations, known as mutations, of a given quiver (i.e., directed graph). Maximal green sequences were introduced by Keller in [13] in order to obtain combinatorial formulas for the refined Donaldson-Thomas invariants of Kontsevich and Soibelman [14]. They are also impor- tant in string theory [1], representation theory [3, 4], and cluster algebras [10].
Recently, there have been many developments on the combinatorics of maximal green sequences (for example, see [11,16,15] and references therein). Our goal is to add to the known combinatorics by developing a numerical invariant of the set of maximal green sequences of Q: the length of minimal length maximal green sequences.
This invariant is natural from the perspective of cluster algebras. Fixing a quiver Q induces an orientation of the edges of the corresponding exchange graph, and the max- imal green sequences of Q are in natural bijection with finite length maximal directed paths in the resulting oriented exchange graph (see [3]). Examples of oriented exchange graphs include the Hasse diagrams of Tamari lattices and Cambrian lattices of type A, D, andE [17]. In these examples the minimal length maximal green sequences always have length equal to the number of vertices ofQ, but in general the minimal length of a maximal green sequence is not known. Thus understanding the length of minimal length maximal green sequences provides new information about oriented exchange graphs.
Our main result (Theorem 5) is a formula for this minimal length whenQis ofcluster type Dn and cluster type Arn´1 (i.e., Q is in the mutation class of a type Dn or of an affine typeAn´1Dynkin quiver). This number was calculated in cluster typeAin [6].
InSection 2, we review the notions of quiver mutation and maximal green sequences.
We also recall some relevant results in cluster typeA. We then state our first main result, which allows one to calculate the length of minimal length maximal green sequences of a quiver with finitely many cluster typeAquivers attached to it (seeTheorem 2).
In Section 3, we recall how triangulations of Riemann surfaces can be used to model maximal green sequences of an important family of quivers. We use this model to present Theorem 4, which is crucial to proving Theorem 2. We state Theorem 4 in restricted generality in order to formulate it in a way that fits with our exposition.
Finally, in Section 4, we combine Theorem 2 and the combinatorics of surface trian- gulations to find the length of minimal length maximal green sequences of quivers of cluster typeDn and cluster typeArn´1.
2 Ice quivers and maximal green sequences
Aquiver Qis a 4-tuple pQ0,Q1,s,tq, whereQ0 “ rms:“ t1, 2, . . . ,mu is a set ofvertices, Q1 is a set of arrows, and two functionss,t : Q1 Ñ Q0 defined so that for everyα P Q1, we have spαqÝÑα tpαq. An ice quiveris a pair pQ,Fqwith Q a quiver and F ĂQ0 a set of frozen vertices where any i,j P Fhave no arrows of Qconnecting them. By convention, we assumeQ0zF“ rnsand F“ rn`1,ms :“ tn`1,n`2, . . . ,mu. We refer to elements of Q0zF asmutable vertices. Any quiverQ is regarded as an ice quiver by settingF “ H.
If an ice quiver pQ,Fq is 2-acyclic (i.e., Q has no loops or 2-cycles), we can define a local transformation of it calledmutation. ThemutationofpQ,Fqat a mutable vertex k, denoted µk, produces a new ice quiverpµkQ,Fqby the three step process:
(1) For every 2-path iÑkÑj inQ, adjoin a new arrow iÑj.
(2) Reverse the direction of all arrows incident to k inQ.
(3) Remove any 2-cycles, and remove any arrows that connect two frozen vertices.
From now on, we will only work with 2-acyclic quivers. We show an example of mutation below with the mutable (respectively, frozen) vertices in black (respectively, blue).
pQ,Fq = 1
2
3 4
;;;; ##;; µ2ÞÝÑ 1
2
3 4
cc ////
{{{{ ;; = pµ2Q,Fq
Let Mut(pQ,Fq) denote the collection of ice quivers obtainable frompQ,Fqby finitely many mutations where such ice quivers are considered up to an isomorphism of quivers that fixes the frozen vertices. We refer to Mut(pQ,Fq) as themutation classofpQ,Fq. The following description of the mutation class of cluster typeAn quivers will be useful.
Lemma 1. [5, Prop. 2.4] A connected quiver Q with n vertices is of cluster typeAn if and only if Q satisfies the following:
iq All non-trivial cycles in the underlying graph of Q are oriented and of length 3.
iiq Any vertex has degree at most 4.
iiiq If a vertex has degree 4, then two of its adjacent arrows belong to one 3-cycle, and the other two belong to another 3-cycle.
ivq If a vertex has degree 3, then two of its adjacent arrows belong to a 3-cycle, and the third arrow does not belong to any 3-cycle.
The framed quiver of Q is the ice quiver Qp where Qp0 :“ Q0\ rn`1, 2ns, F “ rn` 1, 2ns, andQp1 :“Q1\ ti Ñn`i :i P rnsu. A mutable vertexiof some quiverQPMutpQqp isgreen(respectively, red) if there are no arrows in Qof the formi Ðn`j(respectively, iÑn`j) for some j P rns. Sign-coherence ofc-vectors [7, Theorem 1.7] implies that any mutable vertex of a quiverQP MutpQqp is either green or red.
Definition 1 ([13]). Amaximal green sequenceof Q is a sequencei “ pi1, . . . ,ikqof mutable vertices ofQ wherep
iq for all jP rksvertex ij P rnsis green in µij´1 ˝ ¨ ¨ ¨ ˝µi1pQq, andp iiq each mutable vertex i P rnsofµipQqp is red whereµi :“µik˝ ¨ ¨ ¨ ˝µi1.
It is an open problem to determine what positive integers can be realized as lengths of maximal green sequences of a quiver Q. In [3, Lemma 2.20], it is shown that if Q is acyclic, then Q has a maximal green sequence of length #Q0 and there are no shorter maximal green sequences of Q. The following theorem is the first to address this open problem for an infinite family of quivers, many of which are not acyclic.
Theorem 1. [6] The length of a minimal length maximal green sequence of a cluster type An
quiver Q is n`#t3-cycles of Qu.
Our first main result (Theorem 2) shows that the problem of finding the minimal length of maximal green sequences of a quiver reduces to solving this problem for quiv- ers Q without any cluster typeA quivers attached to Q as in the following definition.
The proof ofTheorem 2uses properties of theconsistent scattering diagramassociated with quiver Q, which are proven in [12]. We refrain from discussing scattering diagrams here. Instead, we emphasize the applications ofTheorem 2 in this paper.
Definition 2. Given a quiver Q letQ be a quiver composed of full connected subquivers Q,r Q1, Q2, . . . ,Qk, such that all of the following conditions hold:
‚ Qi0XQ0“ txiu,
‚ Qi0XQj0“
! txiu if xi “xj
∅ otherwise ,
‚ for every arrow inQ, whenever one of the endpoints belongs to Qr i0ztxiuthen the other endpoint belongs to Qi0,
‚ for every i the quiver Qi is of cluster typeA.
Q:r Q
Q1
Q2 Qk
xk
x1
x2
Theorem 2. LetQ be a quiver as inr Definition 2. Then the minimal length of a maximal green sequence ofQ is lr min`l1min`l2min` ¨ ¨ ¨ `lkmin´k where limin (respectively, lmin) is the minimal length of a maximal green sequence for Qi (respectively, Q).
3 Surface triangulations and shear coordinates
Let S denote an oriented Riemann surface, and let M ĂS be a finite subset of Swhere we require that for each component B of BS we have BXM ‰ H. We call the elements ofM marked pointsand the elements ofMzpMX BSqpunctures. We call the pairpS,Mq amarked surface4. We henceforth fix a marked surfacepS,Mq.
We define an arconSto be a curve γinSsuch that
4We require thatpS,Mqis not a sphere with one, two, or three punctures; a disc with one, two, or three marked points on the boundary; or a punctured disc with one marked point on the boundary.
γ δ
ν η ÐÑ
γ0
δ
ν η
Figure 1: A flip connecting two trian- gulations of an annulus.
1 2
3 ù
1 2
CC3
oooo
Figure 2: The (cluster typeAr2) quiver QTdefined by a triangulationT.
‚ its endpoints are marked points;
‚ γdoes not intersect itself, except that its endpoints may coincide;
‚ except for the endpoints,γis disjoint fromMand from the boundary ofS;
‚ γdoes not cut out an unpunctured monogon or and unpunctured digon. (In other words,γis not contractible intoMor onto the boundary of S.)
An arc γ is considered up to isotopy relative to the endpoints of γ. We say two arcs γ1 and γ2 on S arecompatible if they are isotopic relative to their endpoints to curves that are nonintersecting except possibly at their endpoints. A triangulation of pS,Mq, denotedT, is defined to be a maximal collection of pairwise compatible arcs.
One moves between triangulations by local moves called flips. Define the flip of an arc γ P T as the unique arc γ1 ‰ γ that produces a triangulation T1 “ pTztγuq \ tγ1u (seeFigure 1). IfM contains punctures, there exist triangulations containingself-folded triangles(e.g., the region ofSbounded byγ3andγ4inFigure 3is a self-folded triangle).
We refer to the arc γ3(respectively, γ4) inFigure 3as aloop(respectively, a radius).
As the flip of a radius is not defined, tagged arcs were introduced in [8] to obtain such a notion. A tagged arc ιpγq is obtained from an arc γ that does not cut out a once-punctured monogon and “tagging” its ends either asplainornotchedso that:
• an end ofγlying on the boundary ofSis tagged plain; and
• both ends of a loop have the same tagging.
We use the symbol ’ to indicate that an end of an arc is notched. We say two tagged arcs ιpγ1qand ιpγ2q arecompatibleif the following hold:
• Their underlying arcs γ1 and γ2 are the same, and the tagged arcsιpγ1qand ιpγ2q have the same tagging at exactly one endpoint.
• Their underlying arcs γ1 and γ2 are distinct and compatible, and any common endpoints ofιpγ1qand ιpγ2qhave the same tagging.
Atagged triangulation ofpS,Mq is a maximal collection of pairwise compatible tagged arcs. It follows from the construction that any arc in a tagged triangulation can be flipped. For example, seeFigure 3for an example of how to flip the radiusγ4.
γ1
γ2
γ3
γ4
γ1
γ2
γ4
ι
ι(γ3)
γ1
γ2
ι(γ3) flip
././
γ40
Figure 3: The map identifying a triangulation of a punctured disk as a tagged tri- angulation of a punctured disk and the flip of the tagged arc γ4 from that tagged triangulation.
Each triangulation T of S defines a signed adjacency quiver QT by associating ver- tices to arcs and arrows based on oriented adjacencies (see Figure 2). More precisely, given a triangulationTconsider a mapπ : TÑTon the set of arcs defined as follows. If γis a radius of a self-folded triangle, then let πpγqbe the corresponding loop, otherwise let πpγq “ γ. Then the quiver QT consists of vertices iγ for every γ P T and arrows iγ Ñ iγ1 for every non self-folded triangle with sides πpγq and πpγ1q such that πpγq follows πpγ1qin the clockwise order. Finally, remove any 2-cycles from QT to produce a 2-acyclic quiver5. The following theorem shows that flips are compatible with mutations.
Theorem 3. [8] Given a tagged triangulationT and a tagged arcγPT, letT1 be obtained from Tby flippingγ. ThenµiγpQTq “ QT1.
We now show how shear coordinates provide a way to describe maximal green se- quences geometrically. Recall that a fixed orientation O of a surface S induces an ori- entation O on each component of BS such that the surface S lies to the right of every component. If γ is a tagged arc inS, the elementary lamination `γ is a curve that runs along γ within a small neighborhood of it. If γ has an endpoint M on the boundary BS, then `γ begins at a point M1 R M on BS located near M in the direction of O, and proceeds alongγ. Ifγ has an endpoint at a puncture p, then`γ spirals into p: clockwise if γ is notched at p, and counterclockwise if it is tagged plain (e.g., see Figure 4). A set of curves L is a lamination if it consists of elementary laminations arising from some pairwise compatible tagged arcs.
Definition 3. Let L be a lamination, and letTbe a triangulation. For each arcγP T, that is not a radius of a self-folded triangle, theshear coordinate6of L with respect toT, denoted by bγpT,Lq, is defined as a sum of contributions from all intersections of curves in L withγ. Specifically, such an intersection contributes +1 (respectively, ´1) to bγpT,Lq if the corresponding segment of
5For the definition ofQTwhereTis any tagged triangulation, we refer the reader to [8].
6For the definition of shear coordinates whereTis any tagged triangulation, we refer the reader to [9].
O O
’
Figure 4: Lamination corresponding to the given set of arcs.
`1
`2
γ δ
Figure 5: The shear coordinate of γ with respect to`1is`1, and the shear coordinate of δ with respect to `2 is
´1.
a curve in L cuts through the quadrilateral surrounding γ as shown in Figure 5 on the left (respectively, right).
Definition 4. FixT “ tγ1, . . . ,γnu,T1a tagged triangulation obtained fromTby a sequence of flips, andγ1 P T1. Define the c-vectorofγ1to be the integer vector cpγ1,T1q :“ pbγ1pT1,`γjqq P Zn. Defineγ1to begreen(respectively,red) if bγ1pT1,`γjq ě 0(respectively, bγ1pT1,`γjq ď 0) for eachγj PT.
By [9, Proposition 17.3], tagged triangulations and elements of MutpQpTqare in bijec- tion. Thus the ice quiver Q P MutpQpTqcorresponding to T1 has QT1 as its mutable part and its other arrows are given by the equations bγ1pT1,`γjq “ #tiγ1 Ñ n`ju ´#tiγ1 Ð n`ju, one for eachγ1 PT1 andγj PT. Moreover, tagged arcγ1P T1is green (respectively, red) if and only if vertex iγ1 P Q0 is green (respectively, red). Additionally, [4] implies that a maximal green sequence i “ pi1, . . . ,ikq of QT is equivalent to the sequence of c-vectorscpiq:“ pcpγp1q,T1q, . . . ,cpγpkq,Tkqqwhereγpjq is the tagged arc corresponding to vertexij P pµij´1¨ ¨ ¨µi1pQpTqq0 and the mutable part ofµij¨ ¨ ¨µi1pQpTqisQTj.
Theorem 4. Let Q: be a full subquiver of QT whose vertices correspond to γj1, . . . ,γj` and let i “ pi1, . . . ,ikqbe a maximal green sequence of QT. Then the sequence c: obtained by removing fromcpiqeveryc-vector with nonzero entries at positions other than j1, . . . ,j` is the sequence of c-vectors of a maximal green sequence of Q:.
The above is essentially proved in [16], using the consistent scattering diagram ofQT. Moreover, the same technique proves Theorem 4 for a general 2-acyclic quiver Q with the c-vectors ofQ appropriately defined. This more general version of Theorem 4 is an ingredient in the proof ofTheorem 2.
Q: c Q1 a
b
Figure 6: Type I quivers
Q: Q1 c d Q2
a
b
Figure 7: Type II quivers
Q: Q1 c d Q2
a
b
Figure 8: Type III quivers
Q:
Q1
Q2 Qk
a2 a1 b1
a3 b2
ak bk
Figure 9: Type IV quivers
4 Quivers of cluster type D
nand A r
n´1To present our results, we need to recall the classification of cluster typeDn quivers. By [18, Theorem 3.1], the mutation class of a type Dn quiver where n ě 4 consists of four families of quivers. Before presenting these, we say a vertexcofQis aconnectingvertex if it has degree at most 2, and if it has degree 2, it belongs to a 3-cycle of the same quiver.
A quiver Q is ofType I(seeFigure 6) if and only if Qhas the following properties:
• it has a full subquiver of the form a Ø c Ø b (the notation a Ø c to indicates that there exists a single arrow inQconnecting aand c),
• the full subquiver Q1of Qon the vertices Q0zta,buis of cluster typeA, and
• the vertexc is a connecting vertex ofQ1.
A quiver Q is ofType II (seeFigure 7) if and only ifQhas the following properties:
• it has a full subquiver of the form shown inFigure 7whose vertices area,b,c,d,
• the subquiver Q1 obtained by removing vertices a and b and the arrow c Ñ d consists of two cluster typeAquiversQ1and Q2, and
• the vertexc (respectively, d) is a connecting vertex ofQ1(respectively, Q2).
A quiverQis of Type III(seeFigure 8) if and only if Qhas the following properties:
• it has a full subquiver as shown inFigure 8whose vertices are a,b,c,d,
• the full subquiverQ1 ofQon the verticesQ0zta,buconsists of two connected quiv- ers Q1 and Q2, each of which is of cluster typeA, and
• the vertexc (respectively, d) is a connecting vertex ofQ1(respectively, Q2).
A quiverQis of Type IV(seeFigure 8) if and only if Qhas the following properties:
• it has a full subquiverRthat is an orientedk-cycle wherekě3,R0“ ta1,a2, . . . ,aku, and R1 “ tai Ñai`1: iP rk´1su \ tak Ña1u,
• for each arrow α P R1, there may be a vertex bi P Q0zR0 that is in a 3-cycle bi Ñ ai Ñα ai`1 Ñ bi, which is a full subquiver of Q, but there are no other vertices in Q0zR0 that are connected to vertices of R,
• the full subquiver Q1 obtained from Qby removing the vertices and arrows of the subquiver R consists of the quiverstQiuiPrks some of which may be empty quivers, and where each quiverQi is of cluster typeAand hasbi as a connecting vertex.
The cluster type Arn´1 quivers can be described in a very similar way to the Type IV quivers (see [2]). However, the only oriented cycles in cluster typeArn´1 quivers are 3-cycles. Another important distinction betweenDn and Arn´1 quivers is that MutpQqp is finite (respectively, infinite) ifQ is of cluster typeDn (respectively,Arn´1).
Theorem 5. Let l denote the length of a minimal length maximal green sequence of a quiverQ.r i) IfQ is of Type I or is of cluster typer Arn´1, then l“n`#t3-cycles inQu.r
ii) IfQ is of Type II, then lr “n`1`#t3-cycles in Q1u `#t3-cycles in Q2u.
iii) IfQ is of Type III, then lr “n`2`#t3-cycles inQu.r
iv) IfQ is of Type IV, then lr “n`k´2`#tai : degpaiq “4u `řk
i“1#t3-cycles in Qiu.
To prove Theorem 5, first, observe that Qr satisfies Definition 2. One thus applies Theorem 2to reduce the problem of calculating l to calculating the minimal length of a maximal green sequence ofQ, which is obtained by removing all vertices ofQrbelonging toQi0zQr0for somei. IfQris of Type I, II, or III, the resulting family of quivers Qconsists of exactly six quivers. Thus, it is a finite calculation to verify the theorem.
On the other hand, ifQris of Type IV or of cluster typeArn´1, the same family of quiv- ers Qis infinite. However, one can parameterize this family of quivers by triangulations of the once-punctured disk, in the former case, and triangulations of the unpunctured annulus, in the latter case, with certain conditions on the triangles. We then use the corresponding triangulation to construct a minimal length maximal green sequence of each quiver Q. We will sketch this approach when Qis a Type IV quiver.
Let pS,Mq be the once-punctured disk with n marked points on the boundary and unique puncture p. A triangulation Tdefining Q is determined by the following prop- erties (see the top left triangulation in Figure 10):
‚ ifγ is not connected to p, there is a triangle whose only internal arc isγ, and
‚ ifγ is connected to p, it is tagged plain at p.
The triangulation T is the unique triangulation satisfying QT “ Q, up to the action of the mapping class group of the surface and up to simultaneously changing the tagging of all ends of arcs connected to p.
The maximal green sequence we construct has five components, which we denote by i1, i2, i3, i4, and i5. We construct the sequence in the context of Figure 10 wherein
γas,γbt P T are the arcs corresponding to vertices as,bt P QT, respectively. We do not justify it here, but the length ofiisn`k´2`#tai: degpaiq “4u7.
The first components,i1 “ pa7,b6,a4,b3,b2,a2,b1qandi2“ pa3,a5,a6,a1q, perform flips at each of the arcs in the initial triangulation T exactly once so that the total length of these two sequences isn. The vertices inijwith j“1, 2 are ordered so that if two vertices inij are connected by an arrow αP pQTq1, then spαqis mutated beforetpαqinij.
The sequence i3 consists of all green vertices of µi2µi1pQpTqwhere flipping the corre- sponding arcs produces ones that can be obtained from arcs ofT not connected to p by moving their endpoints clockwise along the boundary to the next two marked points. As shown in Figure 10, we have i3 “ piαq. Moreover, the new arc α1 is obtained by moving the endpoints ofγb2 clockwise along the boundary to the next two marked points.
The sequence i4 consists of all vertices of µi3µi2µi1pQpTq whose corresponding tagged arcs are connected to pand is tagged plain at p. Similar toi2, we also require that if two vertices in i4 are connected by an arrow α P µi3µi2µi1pQpTq, then spαq is mutated before tpαqini4. As shown inFigure 10, we havei4“ piδq. It turns out that the length ofi3 plus the length ofi4is #tai : degpaiq “4u.
Lastly, the sequence i5 is defined inductively. First, mutate at all green vertices of µi4µi3µi2µi1pQpTqwhose corresponding tagged arcs appear in a triangle whose other two sides are tagged arcs notched at p. In the context of Figure 10, one mutates at the vertices corresponding to σ1p1q and σ2p1q. Next, repeat this process, each time with the new ice quiver, until there are no remaining green vertices. In the example inFigure 10, this process must be repeated twice: in the second (respectively, third) iteration one mutates at the vertices corresponding to σ1p2q and σ2p2q (respectively, σ1p3q). The length of i5 isk´2.
Acknowledgements
A. Garver received support from an RTG grant DMS-1148634, NSERC, and the Canada Research Chairs program. K. Serhiyenko was supported by the NSF Postdoctoral Fellow- ship MSPRF-1502881. The authors are grateful to the referees for their careful comments.
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